Rational Functions and Holes

A rational function has a hole at x = c if the numerator's zero multiplicity exceeds or equals the denominator's at that input.

Functions30–40% of exam

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Context

What this topic is and why it exists

Rational functions have holes when a factor cancels in both the numerator and the denominator.

This occurs at specific input values where the multiplicity of a zero in the numerator matches or exceeds that in the denominator.

You might think a hole is just another zero, but that's the trap.

A zero affects the output directly, while a hole means the function is undefined at that point, even though it behaves normally nearby.

The graph will approach a point but never actually reach it at the hole.

To find these holes, factor both the numerator and the denominator and cancel common factors.

The input value where the factor equals zero is where the hole occurs.

But remember, once you cancel, the hole doesn't show up in the simplified expression.

Ignoring this can lead to errors when sketching graphs or solving equations.

The challenge is recognizing that a hole represents a missing point, not a break or asymptote.

Misidentifying it can lead to wrong conclusions about the function's behavior locally, especially when you're looking at limits or continuity.

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